A Second Look at the Data

Under the null hypothesis, a z-test statistic is computed on a first batch of n observations, giving Z1. If |Z1| > 1.96 the experimenter stops and rejects. Otherwise a second batch of n observations is added, and the z-statistic Z2 of the pooled 2n observations is computed; the experimenter rejects if |Z2| > 1.96. The observations are i.i.d. standard normal with known variance. What is the overall probability of rejecting a true null? Give your answer rounded to 4 decimal places; 3 decimal places is also accepted.

• distributions
• hypothesis testing
• optimal stopping
• variance

Source: Group-sequential testing; two-look error inflation is a standard calculation (cf. Armitage, McPherson and Rowe, 1969). Statement written for AxiomIQ.